
In a calibration stage, we use variants of ensemble Kalman methods. These approaches were originally developed as a derivative-free method for state estimation [1]}, [2]} and are now widely used in NWP [3]}. The methods were subsequently developed for simultaneous
state and parameter estimation, and variants were developed to deal with strongly nonlinear systems [4]}, [5]}, [6]}, [7]}, [8]}. They were eventually recognized as a general purpose tool for the solution of inverse problems whose objective is parameter estimation [9]}, [10]}, [11]}, [12]}. Here we use an optimization approach, referred to as ensemble Kalman inversion [13]}, which builds on the work of [9]}, [10]}. Ensemble Kalman inversion may be understood as
gradient descent projected to the ensemble subspace [16]};
relatedly, iterated EnKF smoothers may be interpreted as
Gauss-Newton type methods confined to the ensemble subspace [7]}.
However, ensemble Kalman inversion and other ensemble Kalman methods do not provide a basis for systematic uncertainty quantification, except in linear Gaussian problems [18]}, [19]}, [20]}.

In an emulation stage, we train an emulator on the climate model statistics generated during the calibration stage in order to quantify uncertainties. To emulate how the climate model statistics depend on parameters to be calibrated, we use Gaussian processes (GPs), a hierarchical machine learning method that learns smooth functions from a set of noisy training points [21]}, [22]}. The GP approach also learns the statistics of the uncertainty in the resulting predicted function. The training points here are provided by the climate model runs performed in the calibration stage.

In a sampling stage, we approximate the
posterior distribution on parameters given model and data,
using the GP emulator to replace the parameter-to-climate statistics map. The emulator is used to estimate error statistics and to sample from the approximate posterior distribution with MCMC. Because the GP emulator is computationally cheap to evaluate and is smooth by virtue of the smoothing properties of GPs, this avoids the issues that limit the usability of MCMC for sampling from climate models directly.

